My derm prefers polysporin over neo. Neosporin contains neomycin which many people are allergic to. Polysporin has higher concentrations of the other antibacterial ingredients and does NOT contain neomycin. Since you have sensitive skin, I'd definitely recommend polysporin.
If you have problems with bandaids/adhesives, go to any pharmacy (even Target in the first aid aisle) and get 2x2 size gauze pads and use paper tape to secure. You should be able to find that there as well - try to find 1/2" width and put most of it over the gauze and just a small edge on your skin. Most people with adhesive allergies do well with paper tape. This is what it looks like
http://www.amazon.com/Micropore-Pape...7524111&sr=1-1 You can also use 1" width and put the gauze pad in the middle like making a bandaid. I think I have seen paper tape in tan as well as white.
Obviously, keep the wound clean. I would change the dressing twice a day, cleaning the wound with soap and water each time, then applying the polysporin (also available at Target). You could even buy larger gauze pads to use for cleaning because they are sterile. Buy gauze that comes in individual packages, usually 2 pads per package and they'll be in a box. Be sure to wash your hands before cleaning the wound. Do not use antibacterial soap, just plain old soap and water. Of, if you find a cleanser in the first aid aisle that might be good, too. If you want to be extra cautious, wear gloves when you change the dressing. This all might seem like overkill, but I think it's worth a shot to go the extra mile to try to prevent an infection.
Good luck and I agree with trying supplements. Vitamins B and C help with healing and collagen production. What about adding probiotics?